ANALYTICAL MECHANICS ant. Then, solving the' triangle formed by these vectors, we obtain a&cos« (ID and where a, 6, and c are the magnitudes of a, b, and ct respectively, while 0 and 6 are the angles b and c make with a, Equation (II) determines the magnitude and equation (III) the direction of c. FIG. 5. Special Cases, (a) If a and b have the same direction, as in Fig. 6a, then <t> = 0. Therefore and tan 6 = 0, $ » 0. Thus c has the same direction as a and b, while its magnitude equals the arithmetical sum of their magnitudes. (b) When a and b are oppositely directed, as in Fig. Ob, ^ = TT. Therefore and tan 0 = 0, 0 = 0.